This week was a fairly productive one for me working through KA. I FINALLY made it through the Applications of Multivariable Functions unit test which I got through on Tuesday. It turned out not to be too difficult, but I had to do two attempts at it. (I actually got a question wrong on my second attempt at it, but KA has changed the way that it works and as long as you don’t get the same type of question wrong on a subsequent attempt, I get full marks for having got that type of question correct on the previous attempt. Boom.) I started the fourth unit, Integrating Multivariable Functions, on Wednesday and made it through the first section relatively quickly. The following section was composed of five articles and the section after that had nine videos and four exercises. I made it through the first article in the second section and then skipped ahead and watched the first three videos in the third section. (I feel like that may have been a bit hard to follow…) So in total I made it through the unit test, seven videos, one exercise, and one article this week. Some pretty decent progress! PLUS, I finished off Applications of Multivariable Functions before the end of September which was my goal. 💪🏼😤
I screwed up this week and didn’t take any screen shots of the questions on the unit test for some reason. I made a note about one of the questions that I ended up getting wrong but for some reason forgot to take a screen shot of it. I’m not going to explain the question, but here are my notes from it where the first page shows my initial attempt and the second page shows how I actually should have worked through it:
As I mentioned, I started the following unit, Integrating Multivariable Functions, on Wednesday. The first section was titled Line Integrals for Scaler Functions and, for the first time working in MC, as the title would suggest, brought in the use of integrals into MC. The first thing I learned about was how to determine the area of a line on an (x, y) plane that has different z-values along the line. I was able to understand what was being taught and was generally able to memorize how to use the formula to solve these types of questions. I also got a decent grasp on why the formula works as it does, but, as usual, it was the hardest part to understand. Here are some screen shots from the first two videos in the section and my notes from those videos:
The following two videos in the section went through a specific example of how to use the formula. I found this example particularly helpful as not only did it give me an idea of how to use the formula, I also had to review how to use the reverse chain rule with integrals. In the video Sal (who I forgot to mention is doing these videos!) uses u-substitution which I was never really able to understand. (I came up with my own method a long time which you can see in my notes below.) I had to find the integral of cos(t)sin(t)dt and forgot that 1) I could think of cos(t) as being the derivative of sin(t) and 2) that I could consider sin(t) as being inside a function (x)1 and then integrate that outside function to being (x)2/2. (I don’t know if that makes sense, but you’ll see in my notes what I did.) Here are some screen shots from those two videos and my notes:
The questions in the exercise from this section were substantially easier than the question in the example videos just above. It took me a second to understand what was being asked of me initially in the first question, but once I figured it out all the questions were pretty straightforward to work through. Here are three example questions from that exercise:
Question 1
Question 2
Question 3
Finally, I took some notes on the article I read through which I’ve added below. The article reviewed how to use integrals, in general, but specifically when finding the length of a line segment of a multivariable function, specifically where x and y are their own variables. Once again, I’m note 100% if I have this correct but that’s what I believe is going on. Here are my notes:
Like I said, I don’t completely understand what’s going on. Since y = x2 you can think of the function (I think) as a single variable function, f(x) = x2, but I think what this is getting to is that you could also think of it a multivariable function where x and y are their own functions (something like x(t) and y(t)) where you could find the derivative of each function (x’(t) and y’(t)), find out how y’(t) = x’(t) and then put in the values of each into the integral and use algebra to substitute y’(t) with its equivalent x’(t) value to find the length of the multivariable function.
(Again, no clue if what I just wrote makes any sense…)
I’m now 5% of the way through Integrating Multivariable Functions (80/1,600 M.P.). I’ve definitely worked through units that had more Mastery Points in them, but this unit has a TON of videos and articles in it so I’m assuming it’s going to take me awhile to get through. (There’s one section that has two exercises, which would account for 10% of the total M.P., but has THIRTEEN videos, which will count for zero M.P. 😒) My moonshot goal is to get through M.C. before the end of the year but I’m starting to think that’s a bit unlikely. At the very least, I think I should be able to get through this unit in the next three months which is probably a better goal to give myself. If I can somehow make it through this unit before the end of November, I’ll have a shot at getting through the entire subject before the end of the New Year. So far what I’ve worked through has been pretty enjoyable so if it continues that way, there’s a tiny, TINY chance of me getting through calculus by 2024. So tiny you could think of it as dChance. 🤓